Molecular orbital theory: learning to use the theory objectively

TOPICS

sp 3 Hybridization: a scheme for tetrahedral and related species

4.7 Molecular orbital theory: learning to use the theory objectively

The aim of this section is not to establish complete bonding pictures for molecules using MO theory, but rather to develop an objective way of using the MO model to rationa-lize particular features about a molecule. This often involves drawing a partial MO diagram for the molecule in question.

In each example below, the reader should consider the implications of this partial treatment: it can be dangerous because bonding features, other than those upon which one is focusing, are ignored. However, with care and prac-tice, the use of partial MO treatments is extremely valuable as a method of understanding structural and chemical properties in terms of bonding and we shall make use of it later in the book.

-Bonding in CO

₂

The aim in this section is to develop an MO description of the

-bonding in CO₂. Before beginning, we must consider what valence orbitals are unused after
-bonding. The CO₂ molecule belongs to the D_1hpoint group; the z axis is defined to coincide with the C₁axis (structure 4.3). The
-bonding in an XH₂molecule was described in Figure 4.13. A similar picture can be developed for the
-bonding in CO₂, with the difference that the H 1s orbitals in XH₂are replaced by O 2s and 2pz orbitals in CO2. Their overlap with the C 2s and 2pz orbitals leads to the formation of six MOs with
_g or

usymmetry, four occupied and two unoccupied.

(4.3)

After the formation of CO
-interactions, the orbitals remaining are the C and O 2px and 2py orbitals. We now use the ligand group orbital approach to describe the

-bonding in terms of the interactions between the C 2p_x and 2p_y orbitals and the LGOs (derived from O 2p_x and 2p_y orbitals) of an O^{  }O fragment. The LGOs are shown in Figure 4.24. An in-phase combination of 2p orbitals is non-centrosymmetric and has _u symmetry, while an out-of-phase combination is centrosymmetric and has _g symmetry. Only the _uLGOs have the correct symmetry to interact with the C 2p_x and 2p_y orbitals, leaving the _g LGOs as non-bonding MOs in CO₂. After filling the lower-lying
-bonding MOs, there are eight electrons left. These occupy the _uand _gMOs (Figure 4.24). The characters of one _uMO and one _g MO are shown at the top of Figure 4.24; for each degenerate set of MOs, the character of the second _u MO is the same as the first but is orthogonal to it. Each _uMO has delocalized OCO -bonding charac-ter, and the net result of having both _uorbitals occupied is a

-bond order of 1 per CO interaction.

Chapter 4 ^. Molecular orbital theory: learning to use the theory objectively 119

Self-study exercise

Work out a qualitative MO description for the
-bonding in CO₂ and show that this picture is consistent with leaving eight electrons to occupy the -type MOs shown in Figure 4.24.

[NO

₃

]

^

In worked example 4.2, we considered the bonding in [NO3]^ using a VB approach. Three resonance structures (one of which is 4.4) are needed to account for the equivalence of the NO bonds, in which the net bond order per NO bond is 1.33. Molecular orbital theory allows us to represent the NO -system in terms of delocalized interactions.

O N

–O O^–

(4.4)

The [NO₃]^ion has D_3hsymmetry and the z axis is defined to coincide with the C₃axis. The valence orbitals of each N and O atom are 2s and 2p orbitals. The -bonding in [NO₃]^ can be described in terms of the interactions of the N 2p_z orbital with appropriate LGOs of the O₃ fragment. Under D_3h symmetry, the N 2p_z orbital has a₂’’ symmetry (see Table 4.1). The LGOs that can be constructed from O 2p_z orbitals are shown in Figure 4.25 along with their sym-metries; the method of derivation is identical to that for the corresponding LGOs for the F₃ fragment in BF₃

(equations 4.23–4.25). The partial MO diagram shown in Figure 4.25 can be constructed by symmetry-matching of the orbitals. The MOs that result have -bonding (a₂’’), non-bonding (e’’) and -antibonding (a₂’’^) character; the a₂’’ and a₂’’^ MOs are illustrated at the right-hand side of Figure 4.25. Six electrons occupy the a₂’’ and e’’ MOs. This number of electrons can be deduced by considering that of the 24 valence electrons in [NO₃]^, six occupy
-bonding MOs, 12 occupy oxygen-centred MOs with essentially non-bonding character, leaving six electrons for the -type MOs (seeproblem 4.18at the end of chapter).

Molecular orbital theory therefore gives a picture of [NO3]^ in which there is one occupied MO with -character and this is delocalized over all four atoms giving an NO -bond order of

1

3. This is in agreement with the valence bond picture, but it is perhaps easier to visualize the delocalized bonding scheme than the resonance between three contributing forms of the type of structure 4.4. The bonding in the isoelectronic species [CO₃]^2 and [BO₃]^3 (both D_3h) can be treated in a similar manner.

SF

₆

Sulfur hexafluoride (4.5) provides an example of a so-called hypervalent molecule, i.e. one in which the central atom appearsto expand its octet of valence electrons. However, a valence bond picture of the bonding in SF6 involving resonance structures such as 4.6 shows that the S atom obeys the octet rule. A set of resonance structures is needed to rationalize the observed equivalence of the six SF bonds. Other examples of ‘hypervalent’ species of the p-block elements are PF₅, POCl₃, AsF₅ and [SeCl₆]^2. The Fig. 4.24 A partial MO diagram that illustrates the formation of delocalized CO -bonds using the ligand group orbital approach. The CO₂molecule is defined as lying on the z axis. The characters of the _gand _uMOs are shown in the diagrams at the top of the figure.

bonding in each compound can be described within VB theory by a set of resonance structures in which the octet rule is obeyed for each atom (seeSections 14.3and15.3).

S²⁺

F

F F^–

F^– F

F

(4.5) (4.6)

The SF₆molecule, 4.5, belongs to the O_hpoint group, which is one of the cubic point groups. The relationship between the octahedron and cube is shown in Figure 4.26a; the x, y and z axes for the octahedron are defined as being parallel to the edges of the cube. In an octahedral molecule such as SF₆, this means that the x, y and z axes coincide with the SF bonds. Table 4.4 gives part of the O_h character table, and the positions of the rotation axes are shown in Figure 4.26b.

The SF₆ molecule is centrosymmetric, the S atom being on an inversion centre. Using the O_hcharacter table, the valence orbitals of the S atom in SF₆can be classified as follows:

. the 3s orbital has a_1gsymmetry;

. the 3p_x, 3p_y and 3p_z orbitals are degenerate and the orbital set has t1usymmetry.

Fig. 4.25 A qualitative, partial MO diagram to illustrate the formation of a delocalized -system in [NO₃]^; a ligand group orbital approach is used. The characters of the a₂’’ and a₂’’^MOs are shown in the diagrams at the right-hand side of the figure.

Fig. 4.26 (a) An octahedron can be inscribed in a cube; each vertex of the octahedron lies in the middle of a face of the cube.

(b) The diagram shows one of each type of rotation axis of an octahedron. An inversion centre lies at the centre of the octahedron. [Exercise: Work out where the
_hand
_dplanes lie; see Table 4.4.]

Chapter 4 ^. Molecular orbital theory: learning to use the theory objectively 121

Ligand group orbitals for the F₆fragment in SF₆ can be constructed from the F 2s and 2p orbitals. For a qualitative picture of the bonding, we can assume that the sp separa-tion for fluorine is relatively large (see Section 1.13) and, as a consequence, there is negligible sp mixing. Separate sets of LGOs can therefore be formed from the F 2s orbitals and from the F 2p orbitals. Furthermore, the 2p orbitals fall into two classes: those that point towards the S atom (radial orbitals, diagram 4.7) and those that are tangential to the octahedron (diagram 4.8).

(4.7) (4.8)

The SF
-bonds involve the radial 2p orbitals, and there-fore the partial MO diagram that we construct for SF₆ focuses only on these fluorine orbitals. The wavefunctions that describe the LGOs for the F6 fragment in SF6 are derived as follows. We first work out how many of the six radial 2p orbitals are unchanged under each Oh symmetry operation. The following row of characters gives the result:

E 8C₃ 6C₂ 6C₄ 3C₂ (¼C₄²)

i 6S₄ 8S₆ 3
_h 6
_d

6 0 0 2 2 0 0 0 4 2

This same row of characters can be obtained by summing the characters for the A1g, T1uand E_grepresentations in the Oh

character table (Table 4.4). Therefore, the LGOs have a1g, t1u

and e_gsymmetries.

It is now helpful to introduce the concept of a local axis set. When the LGOs for a Y_n group in an XY_n molecule

involve orbitals other than spherically symmetric s orbitals, it is often useful to define the axis set on each Y atom so that the z axis points towards X. Diagram 4.9 illustrates this for the F₆fragment.

F₃ F1

F6

F5

F₂

F₄

z

z

z z z z

(4.9)

Thus, the six radial 3p orbitals that constitute the basis set for the LGOs of the F₆fragment in SF₆can be taken to be six 3p_z orbitals. Let these be labelled ₁– ₆ (numbering as in 4.9). By using the same method as in previous examples in this chapter, we can derive the wavefunctions for the a_1g, t_1u and e_g LGOs (equations 4.26–4.31). These LGOs are represented schematically in Figure 4.27.

ða_1gÞ ¼ 1 ffiffiffi6

p ð 1þ ₂þ ₃þ ₄þ ₅þ ₆Þ ð4:26Þ ðt_1uÞ₁¼ 1

ffiffiffi2

p ð 1 ₆Þ ð4:27Þ

ðt1uÞ2¼ 1 ffiffiffi2

p ð 2 4Þ ð4:28Þ

ðt1uÞ3¼ 1 ffiffiffi2

p ð 3 5Þ ð4:29Þ

ðegÞ1¼ 1 ffiffiffiffiffi

p12ð2 1 2 3 4 5þ 2 6Þ ð4:30Þ ðegÞ2¼¹₂ð 2 3þ 4 5Þ ð4:31Þ The partial MO diagram in Figure 4.28 is constructed by matching the symmetries of the S valence orbitals and the LGOs of the F6fragment. Orbital interactions occur between the a_1gorbitals and between the t_1uorbitals, but the e_gset on the F₆fragment is non-bonding in SF₆.

There are 48 valence electrons in SF₆. These occupy the a_1g, t_1uand e_gMOs shown in Figure 4.28, in addition to 18 Table 4.4 Part of the O_hcharacter table; the complete table is given in Appendix 3.

O_h E 8C₃ 6C₂ 6C₄ 3C₂

(¼ C²₄)

i 6S₄ 8S₆ 3
_h 6
_d

A_1g 1 1 1 1 1 1 1 1 1 1

A_2g 1 1 1 1 1 1 1 1 1 1

E_g 2 1 0 0 2 2 0 1 2 0

T_1g 3 0 1 1 1 3 1 0 1 1

T_2g 3 0 1 1 1 3 1 0 1 1

A_1u 1 1 1 1 1 1 1 1 1 1

A_2u 1 1 1 1 1 1 1 1 1 1

E_u 2 1 0 0 2 2 0 1 2 0

T_1u 3 0 1 1 1 3 1 0 1 1

T_2u 3 0 1 1 1 3 1 0 1 1

MOs that possess mainly fluorine character. The qualitative MO picture of the bonding in SF₆that we have developed is therefore consistent with six equivalent SF bonds. Based on Figure 4.28, the SF bond order is ²/₃ because there are four bonding pairs of electrons for six SF interactions.

Three-centre two-electron interactions

We have already described several examples of bonding pic-tures that involve the delocalization of electrons. In cases such as BF₃ and SF₆, this leads to fractional bond orders.

We now consider two linear XY₂ species in which there is only one occupied MO with YXY bonding character.

This leads to the formation of a three-centre two-electron (3c-2e) bonding interaction.

In a 3c-2e bonding interaction, two electrons occupy a bonding MO which is delocalized over three atomic centres.

The [HF₂]^ ion (seeFigure 9.8) has D_1h symmetry and the z axis coincides with the C₁ axis. The bonding in [HF₂]^can be described in terms of the interactions of the H 1s orbital (
_gsymmetry) with the LGOs of an F^{  }F frag-ment. If we assume a relatively large sp separation for fluorine, then sets of LGOs can be constructed as follows:

. LGOs formed by combinations of the F 2s orbitals;

. LGOs formed by combinations of the F 2p_z orbitals;

. LGOs formed by combinations of the F 2p_x and 2p_y orbitals.

The method of deriving the wavefunctions that describe these LGOs is as before, and the results are summarized schematically at the right-hand side of Figure 4.29. Although the H 1s orbital is of the correct symmetry to interact with either of the F^{  }F
_g LGOs, there is a poor energy match between the H 1s orbital and F^{  }F 2s2s combination.

Thus, the qualitative MO diagram in Figure 4.29 shows the H 1s orbital interacting only with the higher-lying
_gLGO giving rise to
_g and
_g^ MOs, the character of which is shown in the diagrams at the top of Figure 4.29. All other MOs have non-bonding character. Of the nine MOs, eight are fully occupied. Since there is only one MO that has HF bonding character, the bonding in [HF₂]^ can be described in terms of a three-centre two-electron interaction.

The formal bond order for each HF ‘bond’ is¹₂.

Self-study exercise

How many nodal planes does each of the
_gand
_g^MOs shown at the top of Figure 4.29 possess? Where do these lie in relation to the H and F nuclei? From your answers, confirm that the
_gMO contains delocalized FHF bonding character, and that the
_g^ MO has HF antibonding character.

The second example of a linear triatomic with a 3c-2e bond-ing interaction is XeF₂ (D_1h). The bonding is commonly Fig. 4.27 Ligand group orbitals for the F₆fragment in SF₆(O_h). These orbitals only include contributions from the radial 2p orbitals on fluorine (see text).

Fig. 4.28 Qualitative, partial MO diagram for the formation of SF₆using the ligand group orbital approach with a basis set for sulfur that is composed of the 3s and 3p atomic orbitals.

Chapter 4 ^. Molecular orbital theory: learning to use the theory objectively 123

described in terms of the partial MO diagram shown in Figure 4.30. The Xe 5p_z orbital (
_u symmetry) interacts with the combination of F 2p_zorbitals that has
_usymmetry, giving rise to
_u and
_u^ MOs. The combination of F 2p_z orbitals with
_g symmetry becomes a non-bonding MO in XeF₂. There are 22 valence electrons in XeF₂ and all MOs except one (the
_u^ MO) are occupied. The partial MO diagram in Figure 4.30 shows only those MOs derived from p_z orbitals on Xe and F. There is only one MO that has XeF bonding character and therefore the bonding in XeF₂can be described in terms of a 3c-2e interaction.^†

Three-centre two-electron interactions are not restricted to triatomic molecules, as we illustrate in the next section with a bonding analysis of B2H6.

A more advanced problem: B

₂

H

₆

Two common features of boron hydrides (seeSections 12.5 and12.11) are that the B atoms are usually attached to more than three atoms and that bridging H atoms are often pre-sent. Although a valence bond model has been developed

by Lipscomb to deal with the problems of generating localized bonding schemes in boron hydrides,^†the bonding in these compounds is not readily described in terms of VB theory. The structure of B₂H₆ (D_2h symmetry) is shown in Figure 4.31. Features of particular interest are that:

. despite having only one valence electron, each bridging H atom is attached to two B atoms;

. despite having only three valence electrons, each B atom is attached to four H atoms;

. the BH bond distances are not all the same and suggest two types of BH bonding interaction.

Often, B2H6 is described as being electron deficient; it is a dimer of BH3and possesses 12 valence electrons. The forma-tion of the BHB bridges can be envisaged as in structure 4.10. Whereas each terminal BH interaction is taken to be a localized 2c-2e bond, each bridging unit is considered as a 3c-2e bonding interaction. Each half of the 3c-3c-2e interaction is expected to be weaker than a terminal 2c-2e bond and this is consistent with the observed bond distances in Figure Fig. 4.29 A qualitative MO diagram for the formation of [HF₂]^using a ligand group orbital approach. The characters of the
_g and
_g^MOs are shown at the top of the figure.

†In the chemical literature, the bonding in XeF₂is sometimes referred to as a 3c–4e interaction. Since two of the electrons occupy a non-bonding MO, we consider that a 3c-2e interaction description is more meaningful.

†For detailed discussion of the VB model (called styx rules) see:

W.N. Lipscomb (1963) Boron Hydrides, Benjamin, New York; a summary of styx rules and further discussion of the use of MO theory for boron hydrides are given in: C.E. Housecroft (1994) Boranes and Metallaboranes: Structure, Bonding and Reactivity, 2nd edn, Ellis Horwood, Chichester.

4.31. Bonding pictures for B2H6which assume either sp³or sp² hybridized B centres are frequently adopted, but this approach is not entirely satisfactory.

H

B H

H

H B

H H

H B

H B

H H

H H

(4.10)

Although the molecular orbital treatment given below is an oversimplification, it still provides valuable insight into the distribution of electron density in B₂H₆. Using the ligand group orbital approach, we can consider the inter-actions between the pair of bridging H atoms and the residual B₂H₄fragment (Figure 4.32a).

The B₂H₆molecule has D_2h symmetry, and the D_2h char-acter table is given in Table 4.5. The x, y and z axes are defined in Figure 4.32a. The molecule is centrosymmetric, with the centre of symmetry lying midway between the two B atoms. In order to describe the bonding in terms of the interactions of the orbitals of the B₂H₄ and H^{  }H

fragments (Figure 4.32a), we must determine the symmetries of the allowed LGOs. First, we consider the H^{  }H fragment and work out how many H 1s orbitals are left unchanged by each symmetry operation in the D_2hpoint group. The result is as follows:

E C₂ðzÞ C₂ðyÞ C₂ðxÞ i
ðxyÞ
ðxzÞ
ðyzÞ

2 0 0 2 0 2 2 0

This row of characters is produced by adding the rows of characters for the A_gand B_3urepresentations in the D_2h char-acter table. Therefore, the LGOs for the H^{  }H fragment have a_g and b_3usymmetries. Now let the two H 1s orbitals be labelled ₁ and ₂. The wavefunctions for these LGOs are found by considering how ₁ is affected by each sym-metry operation of the D_2hpoint group. The following row of characters gives the result:

E C₂ðzÞ C₂ðyÞ C₂ðxÞ i
ðxyÞ
ðxzÞ
ðyzÞ

1 2 2 1 2 1 1 2

F

Fig. 4.30 A qualitative MO diagram for the formation of XeF₂using a ligand group orbital approach and illustrating the 3c-2e bonding interaction.

177 pm

Terminal H atom (H_term) Bridging H atom

(H_bridge)

B–H_term = 119 pm B–H_bridge = 133 pm

⬔Hterm–B–H_term = 122°

⬔Hbridge–B–H_bridge = 97°

Fig. 4.31 The structure of B₂H₆determined by electron diffraction.

Chapter 4 ^. Molecular orbital theory: learning to use the theory objectively 125

Multiplying each character in the row by the corresponding character in the A_gor B_3urepresentations in the D_2h charac-ter table gives the unnormalized wavefunctions for the LGOs. The normalized wavefunctions are represented by

equations 4.32 and 4.33, and the LGOs are drawn schemati-cally in Figure 4.32b.

ðagÞ ¼ 1 ffiffiffi2

p ð 1þ 2Þ ð4:32Þ

ðb3uÞ ¼ 1 ffiffiffi2

p ð 1 2Þ ð4:33Þ

The same procedure can be used to determine the LGOs of the B₂H₄ fragment. Since the basis set comprises four orbi-tals per B atom and one orbital per H atom, there are 12 LGOs in total. Figure 4.32c shows representations of the six lowest energy LGOs. The higher energy orbitals possess antibonding BH or B^{  }B character. Of those LGOs drawn in Figure 4.32c, three have symmetries that match those of the LGOs of the H^{  }H fragment. In addition to symmetry-matching, we must also look for a good energy match. Of the two a_gLGOs shown in Figure 4.32c, the one Fig. 4.32 (a) The structure of B₂H₆can be broken down into H₂B^{  }BH₂and H^{  }H fragments. (b) The ligand group orbitals (LGOs) for the H^{  }H fragment. (c) The six lowest energy LGOs for the B₂H₄unit; the nodal plane in the b_2uorbital is shown.

Table 4.5 Part of the D_2hcharacter table; the complete table is given in Appendix 3.

D_2h E C₂ðzÞ C₂ðyÞ C₂ðxÞ i
ðxyÞ
ðxzÞ
ðyzÞ

A_g 1 1 1 1 1 1 1 1

B_1g 1 1 1 1 1 1 1 1

B_2g 1 1 1 1 1 1 1 1

B_3g 1 1 1 1 1 1 1 1

A_u 1 1 1 1 1 1 1 1

B_1u 1 1 1 1 1 1 1 1

B_2u 1 1 1 1 1 1 1 1

B_3u 1 1 1 1 1 1 1 1

with the lower energy is composed of B 2s and H 1s character. Although difficult to assess with certainty at a qualitative level, it is reasonable to assume that the energy of this a_g LGO is not well matched to that of the H^{  }H fragment.

We now have the necessary information to construct a qualitative, partial MO diagram for B₂H₆. The diagram in Figure 4.33 focuses on the orbital interactions that lead to the formation of BHB bridging interactions.

Consideration of the number of valence electrons available leads us to deduce that both the bonding MOs will be occu-pied. An important conclusion of the MO model is that the boron–hydrogen bridge character is delocalized over all four atoms of the bridging unit in B2H6. Since there are two such bonding MOs containing four electrons, this result is consistent with the 3c-2e BHB model that we described earlier.

Glossary

The following terms were introduced in this chapter.

Do you know what they mean?

q orbital hybridization

q sp, sp², sp³, sp³d, sp²dand sp³d²hybridization

q ligand group orbital (LGO) approach q basis set of orbitals

q delocalized bonding interaction q symmetry matching of orbitals q energy matching of orbitals q 3c-2e bonding interaction

Further reading

J. Barrett (1991) Understanding Inorganic Chemistry: The Underlying Physical Principles, Ellis Horwood (Simon &

Schuster), New York – Chapters 2 and 4 give a readable introduction to group theory and bonding in polyatomic molecules.

J.K. Burdett (1997) Chemical Bonds, A Dialog, Wiley, New York – An original re´sume´ of modern valence theory pre-sented in the form of a 19th century style dialogue between teacher and pupil.

F.A. Cotton (1990) Chemical Applications of Group Theory, 3rd edn, Wiley, New York – An excellent text that includes the applications of group theory in bonding analyses.

G. Davidson (1991) Group Theory for Chemists, Macmillan, London – Chapter 10 provides a useful discussion and also illustrates the use of group theory.

R.L. DeKock and H.B. Gray (1980) Chemical Structure and Bonding, Benjamin/Cummings, Menlo Park – A readable Fig. 4.33 A qualitative, partial MO diagram showing the formation of the BHB bridging interactions. The BH and BHB bonding character of the a_gMO, and the BHB bonding character of the b_3uMO are shown in the diagrams on the right-hand side; the orientation of the molecule is the same as in the structure at the bottom of the figure.

Chapter 4 ^. Glossary 127

text, treating VB and MO theories and giving examples of the relationship between photoelectron spectra and MO energy levels.

H.B. Gray (1994) Chemical Bonds, University Science Books, California – An introduction to atomic and molecular struc-ture with numerous illustrations.

S.F.A. Kettle (1985) Symmetry and Structure, Wiley, Chichester – An advanced discussion which includes carefully explained applications of group theory.

L. Pauling (1960) The Nature of the Chemical Bond, 3rd edn, Cornell University Press, Ithaca – A classic book dealing with covalent, metallic and hydrogen bonding from the viewpoint of VB theory.

M.J. Winter (1994) Chemical Bonding, Oxford University Press, Oxford – Chapters 5 and 6 give a basic introduction to hybridization and MO theory in polyatomics.

Problems

4.1 (a) State what is meant by the hybridization of atomic orbitals. (b) Why does VB theory sometimes use hybrid orbital rather than atomic orbital basis sets? (c) Show that equations 4.1 and 4.2 correspond to normalized

wavefunctions.

4.2 Figure 4.4 shows the formation of three sp²hybrid orbitals (seeequations 4.3–4.5). (a) Confirm that the

directionalities of the three hybrids are as specified in the figure. (b) Show that equations 4.3 and 4.5 correspond to normalized wavefunctions.

4.3 Use the information given in Figure 4.6b and equations 4.6 to 4.9 to reproduce the directionalities of the four sp³ hybrid orbitals shown in Figure 4.6a.

4.4 (a) Derive a set of diagrams similar to those in Figures 4.2 and 4.4 to describe the formation of sp²d hybrid orbitals. (b) What is the percentage character of each sp²dhybrid orbital in terms of the constituent atomic orbitals?

4.5 Suggest an appropriate hybridization scheme for each central atom: (a) SiF₄; (b) [PdCl₄]^2; (c) NF₃; (d) F₂O; (e) [CoH₅]^4; (f ) [FeH₆]^4; (g) CS₂; (h) BF₃.

4.6 (a) The structures of cis- and trans-N₂F₂were shown in worked example 3.1. Give an appropriate hybridization scheme for the N atoms in each isomer. (b) What hybridization scheme is appropriate for the O atoms in H₂O₂(Figure 1.16)?

4.7 (a) PF₅has D_3hsymmetry. What is its structure? (b) Suggest an appropriate bonding scheme for PF₅within VB theory, giving appropriate resonance structures.

4.8 (a) Draw the structure of [CO₃]^2. (b) If all the CO bond distances are equal, write a set of resonance structures to describe the bonding in [CO₃]^2. (c) Describe the bonding in [CO₃]^2in terms of a hybridization scheme and compare the result with that obtained in part (b).

4.9 (a) Is CO₂linear or bent? (b) What hybridization is appropriate for the C atom? (c) Outline a bonding scheme for CO₂using the hybridization scheme you have suggested. (d) What CO bond order does your scheme imply? (e) Draw a Lewis structure for CO₂. Is this structure consistent with the results you obtained in parts (c) and (d)?

4.10 What is meant by a ligand group orbital?

4.11 VB and MO approaches to the bonding in linear XH₂ (X has 2s and 2p valence atomic orbitals) give pictures in which the XH bonding is localized and delocalized respectively. Explain how this difference arises.

4.12 Table 4.6 gives the results of a Fenske–Hall self-consistent field (SCF) quantum chemical calculation for H₂O using an orbital basis set of the atomic orbitals of O and the LGOs of an H^{  }H fragment. The axis set is as defined in Figure 4.15. (a) Use the data to construct pictorial representations of the MOs of H₂O and confirm that

Table 4.6 Results of a self-consistent field quantum chemical calculation for H₂O using an orbital basis set of the atomic orbitals of the O atom and the ligand group orbitals of an H^{  }H fragment. The axis set is defined in Figure 4.15.

Atomic orbital or ligand group orbital

Percentage character of MOs with the sign of the eigenvector given in parentheses

1 2 3 4 5 6

O 2s 71 (þ) 0 7 () 0 0 22 ()

O 2p_x 0 0 0 100 (þ) 0 0

O 2p_y 0 59 (þ) 0 0 41 () 0

O 2p_z 0 0 85 () 0 0 15 (þ)

H^{  }H LGO(1) 29 (þ) 0 8 (þ) 0 0 63 (þ)

H^{  }H LGO(2) 0 41 () 0 0 59 () 0

Figure 4.15 is consistent with the results of the calculation.

(b) How does MO theory account for the presence of lone pairs in H₂O?

4.13 Refer to Figure 4.17 and the accompanying discussion. (a) Why does the B 2p_zatomic orbital become a non-bonding MO in BH₃? (b) Draw schematic representations of each bonding and antibonding MO in BH₃.

4.14 The diagrams at the right-hand side of Figure 4.19 show three of the MOs in NH₃. Sketch representations of the other four MOs.

4.15 Use a ligand group orbital approach to describe the bonding in [NH₄]^þ. Draw schematic representations of each of the bonding MOs.

4.16 The II bond distance in I₂(gas phase) is 267 pm, in the [I₃]^þion is 268 pm, and in [I₃]^is 290 pm (for the [AsPh₄]^þ salt). (a) Draw Lewis structures for these species. Do these representations account for the variation in bond distance?

(b) Use MO theory to describe the bonding and deduce the II bond order in each species. Are your results consistent with the structural data?

4.17 (a) BCl₃has D_3hsymmetry. Draw the structure of BCl₃ and give values for the bond angles. NCl₃has C_3v symmetry. Is it possible to state the bond angles from this information? (b) Derive the symmetry labels for the atomic orbitals on B in BCl₃and on N in NCl₃.

4.18 Using Figures 4.22, 4.23 and 4.25 to help you, compare the MO pictures of the bonding in BF₃and [NO₃]^. What approximations have you made in your bonding analyses?

4.19 By considering the structures of the following molecules, confirm that the point group assignments are correct:

(a) BH₃, D_3h; (b) NH₃, C_3v; (c) B₂H₆, D_2h. [Hint: use Figure 3.10.]

4.20 In the description of the bonding of B₂H₆, we draw the conclusion that the two bonding MOs in Figure 4.33 have BH bonding character delocalized over the four bridge atoms. (a) What other character do these MOs possess? (b) Does your answer to (a) alter the conclusion that this approximate MO description is consistent with the valence bond idea of there being two 3c-2e bridge bonds?

4.21 In [B₂H₇]^(4.11), each B atom is approximately

tetrahedral. (a) How many valence electrons are present in

the anion? (b) Assume that each B atom is sp³hybridized.

After localization of the three terminal BH bonds per B, what B-centred orbital remains for use in the bridging interaction? (c) Following from your answer to part (b), construct an approximate orbital diagram to show the formation of [B₂H₇]^from two BH₃units and H^. What does this approach tell you about the nature of the BHB bridge?

Overview problems

4.22 (a) What hybridization scheme would be appropriate for the Si atom in SiH₄?

(b) To which point group does SiH₄belong?

(c) Sketch a qualitative MO diagram for the formation of SiH₄from Si and an H₄-fragment. Label all orbitals with appropriate symmetry labels.

4.23 Cyclobutadiene, C₄H₄, is unstable but can be stabilized in complexes such as (C₄H₄)Fe(CO)₃. In such

complexes, C₄H₄is planar and has equal CC bond lengths:

Fe OC

CO CO

(a) After the formation of CH and CC
-bonds in C₄H₄, what orbitals are available for -bonding?

(b) Assuming D_4hsymmetry for C₄H₄, derive the symmetries of the four -MOs. Derive equations for the normalized wavefunctions that describe these MOs, and sketch representations of the four orbitals.

4.24 (a) Draw a set of resonance structures for the hypothetical molecule PH₅, ensuring that P obeys the octet rule in each structure. Assume a structure analogous to that of PF₅.

(b) To what point group does PF₅belong?

(c) Using PH₅as a model compound, use a ligand group orbital approach to describe the bonding in PH₅. Show clearly how you derive the symmetries of both the P atomic orbitals, and the LGOs of the H₅ fragment.

4.25 What hybridization scheme would be appropriate for the C atom in [CO₃]^2? Draw resonance structures to describe the bonding in [CO₃]^2. Figure 4.34 shows representations of three MOs of [CO₃]^2. The MOs in diagrams (a) and (b) in Figure 4.34 are occupied; the MO in diagram (c) is unoccupied. Comment on the characters of these MOs and assign a symmetry label to each orbital.

(4.11)

Chapter 4 ^. Problems 129